Rapid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold. GAGA shows competitive performance in simulated and real-world datasets, including a 30% improvement over SOTA in single-cell population-level trajectory inference.

Advances in medical imaging technologies have enabled the collection of longitudinal images, which involve repeated scanning of the same patients over time, to monitor disease progression. However, predictive modeling of such data remains challenging due to high dimensionality, irregular sampling, and data sparsity. To address these issues, we propose ImageFlowNet, a novel model designed to forecast disease trajectories from initial images while preserving spatial details. ImageFlowNet first learns multiscale joint representation spaces across patients and time points, then optimizes deterministic or stochastic flow fields within these spaces using a position-parameterized neural ODE/SDE framework. The model leverages a UNet architecture to create robust multiscale representations and mitigates data scarcity by combining knowledge from all patients. We provide theoretical insights that support our formulation of ODEs, and motivate our regularizations involving high-level visual features, latent space organization, and trajectory smoothness. We validate ImageFlowNet on three longitudinal medical image datasets depicting progression in geographic atrophy, multiple sclerosis, and glioblastoma, demonstrating its ability to effectively forecast disease progression and outperform existing methods. Our contributions include the development of ImageFlowNet, its theoretical underpinnings, and empirical validation on real-world datasets.

The computational design of novel protein structures has the potential to impact numerous scientific disciplines greatly. Toward this goal, we introduce \foldflow, a series of novel generative models of increasing modeling power based on the flow-matching paradigm over $3\mathrm{D}$ rigid motions---i.e. the group $\mathrm{SE(3)}$---enabling accurate modeling of protein backbones. We first introduce $\text{FoldFlow-Base}$, a simulation-free approach to learning deterministic continuous-time dynamics and matching invariant target distributions on $\mathrm{SE(3)}$. We next accelerate training by incorporating Riemannian optimal transport to create $\text{FoldFlow-OT}$, leading to the construction of both more simple and stable flows. Finally, we design \foldflowsfm, coupling both Riemannian OT and simulation-free training to learn stochastic continuous-time dynamics over $\mathrm{SE(3)}$. Our family of $\text{FoldFlow}$, generative models offers several key advantages over previous approaches to the generative modeling of proteins: they are more stable and faster to train than diffusion-based approaches, and our models enjoy the ability to map any invariant source distribution to any invariant target distribution over $\mathrm{SE(3)}$. Empirically, we validate $\text{FoldFlow}$, on protein backbone generation of up to $300$ amino acids leading to high-quality designable, diverse, and novel samples.

Proteins are essential for almost all biological processes and derive their diverse functions from complex $3 \rm D$ structures, which are in turn determined by their amino acid sequences. In this paper, we exploit the rich biological inductive bias of amino acid sequences and introduce FoldFlow++, a novel sequence-conditioned $\text{SE}(3)$-equivariant flow matching model for protein structure generation. FoldFlow++ presents substantial new architectural features over the previous FoldFlow family of models including a protein large language model to encode sequence, a new multi-modal fusion trunk that combines structure and sequence representations, and a geometric transformer based decoder. To increase diversity and novelty of generated samples -- crucial for de-novo drug design -- we train FoldFlow++ at scale on a new dataset that is an order of magnitude larger than PDB datasets of prior works, containing both known proteins in PDB and high-quality synthetic structures achieved through filtering. We further demonstrate the ability to align FoldFlow++ to arbitrary rewards, e.g. increasing secondary structures diversity, by introducing a Reinforced Finetuning (ReFT) objective. We empirically observe that FoldFlow++ outperforms previous state-of-the-art protein structure-based generative models, improving over RFDiffusion in terms of unconditional generation across all metrics including designability, diversity, and novelty across all protein lengths, as well as exhibiting generalization on the task of equilibrium conformation sampling. Finally, we demonstrate that a fine-tuned FoldFlow++ makes progress on challenging conditional design tasks such as designing scaffolds for the VHH nanobody.

Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data - allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM’s utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs (immune cells).

Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

In modern relational machine learning it is common to encounter large graphs that arise via interactions or similarities between observations in many domains. Further, in many cases the target entities for analysis are actually signals on such graphs. We propose to compare and organize such datasets of graph signals by using an earth mover’s distance (EMD) with a geodesic cost over the underlying graph. Typically, EMD is computed by optimizing over the cost of transporting one probability distribution to another over an underlying metric space. However, this is inefficient when computing the EMD between many signals. Here, we propose an unbalanced graph EMD that efficiently embeds the unbalanced EMD on an underlying graph into an L1space, whose metric we call unbalanced diffusion earth mover’s distance (UDEMD). Next, we show how this gives distances between graph signals that are robust to noise. Finally, we apply this to organizing patients based on clinical notes, embedding cells modeled as signals on a gene graph, and organizing genes modeled as signals over a large cell graph. In each case, we show that UDEMD-based embeddings find accurate distances that are highly efficient compared to other methods.

We present a method called Manifold Interpolating Optimal-Transport Flow (MIOFlow) that learns stochastic, continuous population dynamics from static snapshot samples taken at sporadic timepoints. MIOFlow combines dynamic models, manifold learning, and optimal transport by training neural ordinary differential equations (Neural ODE) to interpolate between static population snapshots as penalized by optimal transport with manifold ground distance. Further, we ensure that the flow follows the geometry by operating in the latent space of an autoencoder that we call a geodesic autoencoder (GAE). In GAE the latent space distance between points is regularized to match a novel multiscale geodesic distance on the data manifold that we define. We show that this method is superior to normalizing flows, Schr\"odinger bridges and other generative models that are designed to flow from noise to data in terms of interpolating between populations. Theoretically, we link these trajectories with dynamic optimal transport. We evaluate our method on simulated data with bifurcations and merges, as well as scRNA-seq data from embryoid body differentiation, and acute myeloid leukemia treatment.

We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover’s Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in {Õ}(n) time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.